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1.23 Complete example

Example.

Evaluate

\int\frac{3x^{2}+7}{(x-1)(x^{2}+9)}\,dx.

We first express the rational function in terms of partial fractions: we want

\frac{3x^{2}+7}{(x-1)(x^{2}+9)}={\frac{A}{x-1}+\frac{Bx+C}{x^{2}+9}}

and hence $A(x^{2}+9)+(Bx+C)(x-1)=3x^{2}+7$ . Evaluating at $x=1$ , we obtain: ${10A=10}$ and hence ${A=1.}$ For $B$ and $C$ the evaluation trick doesn’t work, so we have to equate coefficients. Subtracting $A(x^{2}+9)$ from each side, we get:

(Bx+C)(x-1)={2x^{2}-2=}\,{(2x+2)(x-1)\;\;\Rightarrow B=C=2.}